pdf file for the current version (2.21)
These are the notes for a course taught at the University of
Michigan in 1989 and 1998. In comparison with my book, the
emphasis is on heuristic arguments rather than formal proofs
and on varieties rather than schemes. The notes also discuss
the proof of the Weil conjectures (Grothendieck and
The word étale.
- Etale Morphisms
- The Etale Fundamental Group
- The Local Ring for the Etale Topology
- Sheaves for the Etale Topology
- The Category of Sheaves on Xet.
- Direct and Inverse Images of Sheaves.
- Cohomology: Definition and the Basic Properties
- Cech Cohomology
- Principal Homogeneous Spaces and H1.
- Higher Direct Images; the Leray Spectral Sequence
- The Weil-Divisor Exact Sequence and the Cohomology of Gm
- The Cohomology of Curves
- Cohomological Dimension.
- Purity; the Gysin Sequence.
- The Proper Base Change Theorem.
- Cohomology Groups with Compact Support.
- Finiteness Theorems; Sheaves of Zl-modules
- The Smooth Base Change Theorem.
- The Comparison Theorem.
- The Kunneth Formula.
- The Cycle Map; Chern Classes
- Poincare Duality
- Lefschetz Fixed-Point Formula.
- The Weil Conjectures.
- Proof of the Weil Conjectures, except for the Riemann
- Preliminary Reductions
- The Lefschetz Fixed Point Formula for Nonconstant Sheaves
- The MAIN Lemma
- The Geometry of Lefschetz Pencils
- The Cohomology of Lefschetz Pencils
- Completion of the Proof of the Weil Conjectures.
- The Geometry of Estimates
There are two different words in French, "étaler", which means spread out or displayed and is used in
"éspace étalé", and "étale", which is rare except in poetry.
According to Illusie, it is the second that Grothendieck chose for étale morphism.
The Petit Larousse defines "mer étale" as "mer qui ne monte ni ne descend", i.e., the sea at the point of high or low
tide. For example, there is the quote from Hugo which I included in my book
"La mer était étale, mais le reflux commencait a se sentir".
I think Grothendieck chose the word because the way he pictured étale
morphisms reminded him of a calm sea at high tide under a full moon
(locally almost parallel bands of light, but not globally). I find
this image beautiful. A footnote in Mumford's Red Book on Algebraic
Geometry says: "The word apparently refers to the appearance of the
sea at high tide under a full moon in certain types of weather."
v2.01; August 9, 1998; first version on the web; 190 pages.
v2.10; May 20, 2008; corrected errors and improved the TeX; 196 pages.
old version 2.10
v2.20; May 3, 2012; corrected; minor improvements; 202 pages.
v2.21; March 22, 2013; corrected; minor improvements; 202 pages.